Chapter 15

• Definition of risk management: the practice of defining the risk level a firm desires, identifying the risk level it currently has, and using derivatives or other financial instruments to adjust the actual risk level to the desired risk level.

Why Practice Risk Management?

• The Impetus for Risk Management– Firms practice risk management for several reasons:

• Interest rates, exchange rates and stock prices are more volatile today than in the past.

• Significant losses incurred by firms that did not practice risk management

• Improvements in information technology

• Favorable regulatory environment

– Sometimes we call this activity financial risk management.

Why Practice Risk Management? (continued)

• The Benefits of Risk Management– What are the benefits of risk management, in light of

the Modigliani-Miller principle that corporate financial decisions provide no value because shareholders can execute these transactions themselves?

• Firms can practice risk management more effectively.

• There may tax advantages from the progressive tax system.

• Risk management reduces bankruptcy costs.

• Managers are trying to reduce their own risk.

Why Practice Risk Management? (continued)

• The Benefits of Risk Management (continued)– By protecting a firm’s cash flow, it increases the

likelihood that the firm will generate enough cash to allow it to engage in profitable investments.

– Some firms use risk management as an excuse to speculate.

– Some firms believe that there are arbitrage opportunities in the financial markets.

– Note: The desire to lower risk is not a sufficient reason to practice risk management.

Managing Market Risk• Market risk: the uncertainty associated with

interest rates, foreign exchange rates, stock prices, or commodity prices.– Example: A dealer with the following positions:

• A four-year interest rate swap with $10 million notional principal in which it pays a fixed rate and receives a floating rate.

• A 3-year interest rate call with $8 million notional principal. The dealer is short and the exercise rate is 12%.

– See Table 15.1, p. 546 for current term structure and forward rates. We obtain the call price as $73,745 and the swap rate is 11.85%.

Managing Market Risk (continued)

• Delta Hedging– We estimate the delta by repricing the swap and

option with a one basis point move in all spot rates and average the price change.

• See Table 15.2, p. 547 for estimated swap and option deltas.

– We are long the swap so we have a delta of $2,130.5, round to $2,131.

– We are short the option so we have a delta of -$244.– Our overall delta is $1,887.

Managing Market Risk (continued)

• Delta Hedging (continued)– We need a Eurodollar futures position that gains $1,887

if rates move down and loses that amount if rates move up. Thus, we require a long position of $1,887/$25 = 75.48 contracts. Round to 75. Overall delta:

• $2,131 (from swap)

• -$244 (from option)

• 75(-$25) (from futures)

• = $12 (overall)

Managing Market Risk (continued)

• Gamma Hedging – Here we deal with the risk of large price moves, which

are not fully captured by the delta.– See Table 15.3, p. 549 for the estimation of swap and

option gammas. Swap gamma is -$12,500, and option gamma is $5,000. Being short the option, the total gamma is -$17,500.

– Eurodollar futures have zero gamma so we must add another option position to offset the gamma. We assume the availability of a one-year call with delta of $43 and gamma of $2,500.

Managing Market Risk (continued)

• Gamma Hedging (continued)– We use x1 Eurodollar futures and x2 of the one-

year calls. The swap and option have a delta of $1,887 and gamma of -$17,500. We solve the following equations:

• $1,887 + x1(-$25) + x2($43) = $0 (zero delta)• -$17,500 + x1($0) + x2($2,500) = $0 (zero gamma)• Solving these gives x1 = 87.52 (go long 88

Eurodollar futures) and x2 = 7 (go long 7 times $1,000,000 notional principal of one-year option)

Managing Market Risk (continued)

• Vega Hedging– Swaps, futures, and FRAs do not have vegas. – We estimate the vegas of the options

• On our 3-year option, if volatility increases (decreases) by .01, option will increase (decrease) by $42 (-$42). Average is $42. We are short this option, so vega = -$42.

• One-year option has estimated vega of $3.50. • Overall portfolio has vega of ($3.50)(7 million) -

$42 = -$17.50.

Managing Market Risk (continued)

• Vega Hedging (continued)– We add a Eurodollar futures option, which has delta of

-$12.75, gamma of -$500, and vega of $2.50 per $1MM.

– Solve the following equations• $1,887 + x1(-$25) + x2($43) + x3(-$12.75) = 0 (delta)

• -$17,500 + x1($0) + x2($2,500) + x3(-$500) = 0 (gamma)

• -$42 + x1($0) + x2($3.50) + x3($2.50) = 0 (vega)

– The coefficients are the multiples of $1,000,000 notional principal we need.

– Solutions are x1 = 86.61, x2 = 8.09375, x3 = 5.46875.

Managing Market Risk (continued)

• Vega Hedging (continued)– Any type of hedge (delta, delta-gamma, or

delta-gamma-vega) must be periodically adjusted.

– Virtually impossible to have a perfect hedge.

How much can we lose?Everything

correct, but useless answer.

How much can we lose realistically?

Value At Risk

• A technique which uses the statistical analysis of historical market trends and volatilities to estimate the likelihood that a given portfolio's losses will exceed a certain amount.

VaR (Value-at-Risk)

It is usually quoted in terms of a time horizon, and a confidence level.

For example, the 10 day 95% VaR is the size of loss X that will not happen 95% of the time over the next 10 days.

5%

95%

(Profit/Loss Distribution)

XValue-at-Risk

The Question Being Asked in VaR

“What loss level is such that we are X% confident it will not be exceeded in N business days?”

VaR and Regulatory Capital

• Regulators base the capital they require banks to keep on VaR

• The market-risk capital is k times the 10-day 99% VaR where k is at least 3.0

VaR vs. C-VaR

• VaR is the loss level that will not be exceeded with a specified probability

• C-VaR is the expected loss given that the loss is greater than the VaR level

• Although C-VaR is theoretically more appealing, it is not widely used

Advantages of VaR

• It captures an important aspect of risk

in a single number

• It is easy to understand

• It asks the simple question: “How bad can things get?”

Value-at-Risk (VAR)– A dollar measure of the minimum loss that would be

expected over a given time with a given probability. Example:

• VAR of $1 million for one day at .05 means that the firm could expect to lose at least $1 million over a one day period 5% of the time.

– Widely used by dealers and increasingly by end users.– See Table 15.4, p. 553 for example of discrete

probability distribution of change in value. VAR at 5 % is $3 million loss.

– See Figure 15.1, p. 554 for continuous distribution.

Value-at-Risk (VAR)

– VAR calculations require use of formulas for expected return and standard deviation of a portfolio:

– where • E(R1), E(R2) = expected returns of assets 1 and 2

1, 2 = standard deviations of assets 1 and 2

= correlation between assets 1 and 2

• w1, w2 = % of one’s wealth invested in asset 1 or 2

ρσσw2wσwσwσ

)E(Rw)E(Rw)E(R

212122

22

21

21p

2211p

Value-at-Risk (VAR)

– Three methods of estimating VAR• Analytical method: Uses knowledge of the

parameters (expected return and standard deviation) of the probability distribution and assumes a normal distribution.

– Example: $20 million of S&P 500 with expected return of .12 and volatility of .15 and $12 million of Nikkei 300 with expected return of .105 and volatility of .18. Correlation is .55. Using the above formulas, the overall portfolio expected return is .1144 and volatility is .1425.

• Value-at-Risk (VAR) (continued)– For a weekly VAR, convert these to weekly figures.

Expected return = .1144/52 = .0022

Volatility = .1425/52 = .0198.

– With a normal distribution, we have

VAR = .0022 - 1.65(.0198) = -.0305

– So the VAR is $32,000,000(.0305) = $976,000.

• Value-at-Risk (VAR) (continued)– Example using options: 200 short 12-month calls on S&P

500, which has volatility of .15 and price of $14.21. Total value of $1,421,000.

» Based on monthly data, expected return is .0095 and volatility is .0412.

» Upside 5 % is .0095 + 1.65(.0412) = .0775, which is 720(1.0775) = 775.80.

» Option would be worth 775.80 - 720 = 55.80 so loss is 55.80 - 14.21= 41.59 per option.

» Total loss = 200(500)(41.59) = $4.159 million. This is the VAR.

Value-at-Risk (VAR) (continued)

– One assumption often made is that the expected return is zero. This is not likely to be true.

– Sometimes rather than use the precise option price from a model, a delta is used to estimate the price. This makes the analytical method be sometimes called the delta-normal method.

– Volatility and correlation information is necessary. See the web site www.riskmetrics.com where data of this sort are provided free.

• Value-at-Risk (VAR) (continued)• Historical method: Uses historical information on the

user’s portfolio to obtain the distribution.

– Example: See Figure 15.2, p. 557. For portfolio of $15 million, VAR at 5% is approximately a loss of 10% or $15,000,000(.10) = -$1,500,000.

– Historical method is subject to limitation that the past holdings of the portfolio may not have the same distributional properties as the future holdings. It also is limited by the results of the chosen time period, which might not be representative of the future.

• Value-at-Risk (VAR) (continued)• Monte Carlo Simulation method: Uses Monte Carlo

method, as described in Appendix 14.B, to generate random outcomes on the portfolio’s components.

• Value-at-Risk (VAR) (continued)– A Comprehensive Calculation of VAR

• We do an example of a portfolio of $25 million in the S&P 500. We want a 5% 1-day VAR using each method. We collect a sample of daily returns on the S&P 500 for the past year and obtain the following parameter estimates: Average daily return = .0457% and daily standard deviation = 1.3327%. These result in annual figures of

2120.2533327.1

1156.)253(0457.0

• Value-at-Risk (VAR) (continued)– A Comprehensive Calculation of VAR (continued)

• Analytical method: We have 0.0457% - (1.65)1.3327% = -2.1533%. So the VAR is

– .021533($25,000,000) = $538,325– The .21 standard deviation is historically a bit high. Re-estimating

with a standard deviation of .15 gives us a daily standard deviation of 0.9430. Then we obtain 0.0474% - 1.65(0.9430) = -1.5086% and a VAR of

– .015086($25,000,000) = $377,150– Are our data normally distributed? Observe Figure 15.3, p. 559.

• Value-at-Risk (VAR) (continued)– A Comprehensive Calculation of VAR (continued)

• Historical method: Here we rank the returns from worst to best. For 253 returns we obtain the 5% worst by observing the .05(253) = 12.65 worst return. We shall make it the 13th worst. This would be -2.0969%. Thus, the VAR is

– .020969($25,000,000) = $524,225

• Value-at-Risk (VAR) (continued)– A Comprehensive Calculation of VAR (continued)

• Monte Carlo simulation method: We shall use an expected return of 12%, a standard deviation of 15% and a normal distribution.

– We generate 253 random returns (this number is arbitrary and should actually be much larger) by the following method:

– where is a standard normal random number.

1/253*deviation) (Standard

53)Return(1/2 (Expected Return

Value-at-Risk (VAR) (continued)A Comprehensive Calculation of VAR

(continued)

• Monte Carlo simulation method (continued): We do this 253 times, rank the returns from worst to best and obtain the 13th worst return, which is -1.3942%. Then the VAR is

– .013942($25,000,000) = $348,550

• Value-at-Risk (VAR) (continued)– A Comprehensive Calculation of VAR

(continued)• So VAR is estimated to be either

– $538,325– $377,150– $524,225– $348,550

• Key considerations: wide ranges such as this are common, real-world portfolios are more complicated than this, ex post evaluation should be done

• Value-at-Risk (VAR) (continued)– Benefits and Criticisms of VAR

• Widely used

• Facilitates communication with senior management

• Acceptable in banking regulation

• Used to allocate capital within firms

• Used in performance evaluation

• Value-at-Risk (VAR) (continued)– Extensions of VAR

• Stress testing• Conditional VAR: expected loss, given that loss exceeds VAR. In

S&P example, this is $719,450.• Cash Flow at Risk (CAR or CFAR), which is more appropriate for

firms with assets that generate cash but for which a market value cannot easily be assigned.

– Example: expected cash flow of $10 million with standard deviation of $2 million. CAR at .05 would be 1.65($2 million) = $3.3 million because

Prob[$10 million - $3.3 million > Actual Cash Flow] = .05.» Note: CAR is a shortfall measure.

• Earnings at Risk (EAR): measures earnings shortfall.

Managing Credit Risk

• Credit risk or default risk is the risk that the counterparty will not pay off in a financial transaction.

• Credit ratings are widely used to assess credit risk.

Netting

– Netting: several similar processes in which the amount of cash owed by one party to the other is reduced by the amount owed by the latter to the former.

• Bilateral netting: netting between two parties.

• Multilateral netting: netting between more than two parties; essentially the same as a clearinghouse.

• Payment netting: Only the net amount of a payment owed from one party to the other is paid.

• Cross-product netting: payments from one type of transaction are netted against payments for another type of transaction.

Netting• Netting by novation: net value of two parties’ mutual

obligations is replaced by a single transaction; often used in FX markets.

• Closeout netting: netting in the event of default, where all transactions between two parties are netted against each other; see example in text.

• The OTC derivatives market has an excellent record of default. Note the Hammersmith and Fulham default where it was found that a town had no legal authority to engage in swaps. The town was able to get out of paying its losses.